Subgroups — C₁₃×#23505 #15709844

Order 3328.

2134 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#418	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₃×#418	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2296	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2041	C₂	1	8	yes	maximal, complemented
6	C₁₃×#1728	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₁₃	—	2	2	no	—
8	C₂³×D₄×C₁₃	C₂²	1	2	yes	—
9	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
10	C₂×C₁₃×#127	—	2	8	no	—
11	C₂×C₁₃×#127	—	2	2	no	—
12	C₂×C₁₃×#127	C₂²	1	4	yes	—
13	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#418	—	2	8	no	—
17	C₁₃×#418	—	2	8	no	—
18	C₂×C₁₃×#102	C₂²	1	8	yes	complemented
19	C₂×C₁₃×#102	C₂²	1	8	yes	complemented
20	C₁₃×#389	C₂²	1	4	yes	complemented
21	C₁₃×#385	—	2	2	no	—
22	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
23	C₁₃×#336	C₂²	1	4	yes	complemented
24	C₂⁵×C₁₃	—	2	2	no	—
25	C₂²×D₄×C₁₃	—	2	16	no	—
26	C₂²×D₄×C₁₃	—	2	4	no	—
27	C₂²×D₄×C₁₃	—	2	2	no	—
28	C₂²×D₄×C₁₃	—	2	4	no	—
29	C₂²×D₄×C₁₃	—	2	8	no	—
30	C₂²×D₄×C₁₃	—	2	4	no	—
31	C₂²×D₄×C₁₃	—	2	4	no	—
32	C₂²×D₄×C₁₃	—	2	4	no	—
33	C₂²×D₄×C₁₃	—	4	16	no	—
34	C₂²×D₄×C₁₃	D₄	1	8	yes	—
35	C₂²×D₄×C₁₃	C₂³	1	4	yes	—
36	C₂³×C₄×C₁₃	D₄	1	2	yes	—
37	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₁₃×M₁₆	—	2	4	no	—
39	C₂×C₁₃×M₁₆	—	2	4	no	—
40	C₂×C₁₃×M₁₆	—	2	4	no	—
41	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
42	C₁₃×#127	—	2	16	no	—
43	C₁₃×#127	—	2	16	no	—
44	C₁₃×#117	—	2	2	no	—
45	C₁₃×#117	—	2	4	no	—
46	C₁₃×#117	D₄	1	4	yes	complemented
47	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
51	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
52	C₂×C₄²×C₁₃	—	2	1	no	—
53	C₂×C₄²×C₁₃	—	2	4	no	—
54	C₂×C₄²×C₁₃	—	2	1	no	—
55	C₂×C₄²×C₁₃	D₄	1	2	yes	—
56	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
57	C₁₃×#102	—	2	16	no	—
58	C₁₃×#102	—	2	16	no	—
59	C₁₃×#95	—	2	4	no	—
60	#23505	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂⁴×C₁₃	—	2	8	no	—
62	C₂⁴×C₁₃	—	2	4	no	—
63	C₂⁴×C₁₃	—	4	16	no	—
64	C₂⁴×C₁₃	—	4	4	no	—
65	C₂⁴×C₁₃	—	4	4	no	—
66	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×D₄×C₁₃	—	2	32	no	—
68	C₂×D₄×C₁₃	—	2	8	no	—
69	C₂×D₄×C₁₃	—	2	16	no	—
70	C₂×D₄×C₁₃	—	2	16	no	—
71	C₂×D₄×C₁₃	—	2	4	no	—
72	C₂×D₄×C₁₃	—	2	4	no	—
73	C₂×D₄×C₁₃	—	2	4	no	—
74	C₂×D₄×C₁₃	—	2	4	no	—
75	C₂×D₄×C₁₃	—	2	4	no	—
76	C₂×D₄×C₁₃	—	4	64	no	—
77	C₂×D₄×C₁₃	—	4	16	no	—
78	C₂×D₄×C₁₃	—	4	8	no	—
79	C₂×D₄×C₁₃	—	4	4	no	—
80	C₂×D₄×C₁₃	—	4	8	no	—
81	C₂×D₄×C₁₃	—	4	4	no	—
82	C₂×D₄×C₁₃	—	4	16	no	—
83	C₂×D₄×C₁₃	C₂×D₄	1	8	yes	—
84	C₂²×C₄×C₁₃	—	2	4	no	—
85	C₂²×C₄×C₁₃	—	2	8	no	—
86	C₂²×C₄×C₁₃	—	2	2	no	—
87	C₂²×C₄×C₁₃	—	2	2	no	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₃	#41	1	4	yes	—
93	C₂²×C₄×C₁₃	#41	1	2	yes	—
94	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₁₃×M₁₆	—	4	8	no	—
96	C₂×C₈×C₁₃	—	2	4	no	—
97	C₂×C₈×C₁₃	—	2	4	no	—
98	C₂×C₈×C₁₃	—	2	4	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	2	4	no	—
101	C₁₃×C₄⋊C₄	—	4	8	no	—
102	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	complemented
103	C₁₃×C₂²⋊C₄	—	2	4	no	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₄²×C₁₃	—	2	4	no	—
106	C₄²×C₁₃	—	2	4	no	—
107	C₄²×C₁₃	—	2	4	no	—
108	C₄²×C₁₃	—	4	2	no	—
109	C₂²×#127	C₂×C₁₃	1	1	yes	characteristic
110	C₂×#418	C₂×C₁₃	1	2	yes	complemented
111	C₂×#418	C₂×C₁₃	1	2	yes	complemented
112	#2296	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#2041	C₂×C₁₃	1	8	yes	complemented
114	#1728	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂³×C₁₃	—	2	4	no	—
116	C₂³×C₁₃	—	2	8	no	—
117	C₂³×C₁₃	—	4	32	no	—
118	C₂³×C₁₃	—	4	4	no	—
119	C₂³×C₁₃	—	4	4	no	—
120	C₂³×C₁₃	—	4	16	no	—
121	C₂³×C₁₃	—	4	16	no	—
122	C₂³×C₁₃	—	8	16	no	—
123	C₂³×C₁₃	—	8	16	no	—
124	C₂³×C₁₃	#120	1	2	yes	—
125	C₂³×C₁₃	#121	1	2	yes	—
126	C₂³×C₁₃	#124	1	2	yes	—
127	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
128	D₄×C₁₃	—	2	8	no	—
129	D₄×C₁₃	—	2	8	no	—
130	D₄×C₁₃	—	4	64	no	—
131	D₄×C₁₃	—	4	4	no	—
132	D₄×C₁₃	—	4	4	no	—
133	D₄×C₁₃	—	4	16	no	—
134	C₂×C₄×C₁₃	—	2	16	no	—
135	C₂×C₄×C₁₃	—	2	4	no	—
136	C₂×C₄×C₁₃	—	2	4	no	—
137	C₂×C₄×C₁₃	—	2	4	no	—
138	C₂×C₄×C₁₃	—	4	4	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	4	no	—
141	C₂×C₄×C₁₃	#120	1	2	yes	—
142	C₂×C₄×C₁₃	#121	1	8	yes	—
143	C₂×C₄×C₁₃	#121	1	2	yes	—
144	C₂×C₄×C₁₃	#123	1	8	yes	—
145	C₂×C₄×C₁₃	#124	1	2	yes	—
146	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	8	yes	—
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
149	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
150	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₈×C₁₃	—	4	4	no	—
152	C₂³×D₄	—	2	2	no	—
153	C₂³×D₄	C₂²×C₁₃	1	2	yes	—
154	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
155	C₂×#127	—	2	8	no	—
156	C₂×#127	—	2	2	no	—
157	C₂×#127	C₂²×C₁₃	1	4	yes	—
158	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
160	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
161	#418	—	2	8	no	—
162	#418	—	2	8	no	—
163	C₂×#102	C₂²×C₁₃	1	8	yes	complemented
164	C₂×#102	C₂²×C₁₃	1	8	yes	complemented
165	#389	C₂²×C₁₃	1	4	yes	complemented
166	#385	—	2	2	no	—
167	C₄×C₄⋊C₄	—	2	2	no	—
168	#336	C₂²×C₁₃	1	4	yes	complemented
169	C₂²×C₁₃	—	2	4	no	—
170	C₂²×C₁₃	—	2	4	no	—
171	C₂²×C₁₃	—	2	4	no	—
172	C₂²×C₁₃	—	4	8	no	—
173	C₂²×C₁₃	—	4	16	no	—
174	C₂²×C₁₃	—	8	32	no	—
175	C₂²×C₁₃	—	8	8	no	—
176	C₂²×C₁₃	—	8	8	no	—
177	C₂²×C₁₃	#394	1	4	yes	—
178	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
179	C₂²×C₁₃	C₂×#121	1	1	yes	characteristic
180	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
181	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
182	C₄×C₁₃	—	2	8	no	—
183	C₄×C₁₃	—	4	4	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	#394	1	4	yes	—
186	C₄×C₁₃	C₂×#121	1	2	yes	—
187	C₄×C₁₃	C₂×#123	1	2	yes	—
188	C₂⁵	—	2	2	no	—
189	C₂²×D₄	—	2	16	no	—
190	C₂²×D₄	—	2	4	no	—
191	C₂²×D₄	—	2	2	no	—
192	C₂²×D₄	—	2	4	no	—
193	C₂²×D₄	—	2	8	no	—
194	C₂²×D₄	—	2	4	no	—
195	C₂²×D₄	—	2	4	no	—
196	C₂²×D₄	—	2	4	no	—
197	C₂²×D₄	—	4	16	no	—
198	C₂²×D₄	D₄×C₁₃	1	8	yes	—
199	C₂²×D₄	C₂³×C₁₃	1	4	yes	—
200	C₂³×C₄	D₄×C₁₃	1	2	yes	—
201	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
202	C₂×M₁₆	—	2	4	no	—
203	C₂×M₁₆	—	2	4	no	—
204	C₂×M₁₆	—	2	4	no	—
205	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
206	#127	—	2	16	no	—
207	#127	—	2	16	no	—
208	#117	—	2	2	no	—
209	#117	—	2	4	no	—
210	#117	D₄×C₁₃	1	4	yes	complemented
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
214	C₂×C₂²⋊C₄	—	2	1	no	—
215	C₂×C₂²⋊C₄	—	2	1	no	—
216	C₂×C₄²	—	2	1	no	—
217	C₂×C₄²	—	2	4	no	—
218	C₂×C₄²	—	2	1	no	—
219	C₂×C₄²	D₄×C₁₃	1	2	yes	—
220	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
221	#102	—	2	16	no	—
222	#102	—	2	16	no	—
223	#95	—	2	4	no	—
224	C₂×C₁₃	—	2	4	no	—
225	C₂×C₁₃	—	8	8	no	—
226	C₂×C₁₃	C₂×#394	1	1	yes	characteristic
227	C₂×C₁₃	#3003	1	2	yes	—
228	C₂×C₁₃	#3038	1	2	yes	—
229	C₂×C₁₃	#3138	1	2	yes	—
230	C₂⁴	—	2	8	no	—
231	C₂⁴	—	2	4	no	—
232	C₂⁴	—	4	16	no	—
233	C₂⁴	—	4	4	no	—
234	C₂⁴	—	4	4	no	—
235	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₂×D₄	—	2	32	no	—
237	C₂×D₄	—	2	8	no	—
238	C₂×D₄	—	2	16	no	—
239	C₂×D₄	—	2	16	no	—
240	C₂×D₄	—	2	4	no	—
241	C₂×D₄	—	2	4	no	—
242	C₂×D₄	—	2	4	no	—
243	C₂×D₄	—	2	4	no	—
244	C₂×D₄	—	2	4	no	—
245	C₂×D₄	—	4	64	no	—
246	C₂×D₄	—	4	16	no	—
247	C₂×D₄	—	4	4	no	—
248	C₂×D₄	—	4	8	no	—
249	C₂×D₄	—	4	8	no	—
250	C₂×D₄	—	4	4	no	—
251	C₂×D₄	—	4	16	no	—
252	C₂×D₄	C₂×D₄×C₁₃	1	8	yes	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	8	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
258	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
260	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
261	C₂²×C₄	C₁₃×#41	1	4	yes	—
262	C₂²×C₄	C₁₃×#41	1	2	yes	—
263	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
264	M₁₆	—	4	8	no	—
265	C₂×C₈	—	2	4	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	4	8	no	—
271	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	complemented
272	C₂²⋊C₄	—	2	4	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₄²	—	2	4	no	—
275	C₄²	—	2	4	no	—
276	C₄²	—	2	4	no	—
277	C₄²	—	4	2	no	—
278	C₁₃	#23505	1	1	yes	characteristic, direct product factor, Sylow
279	C₂³	—	2	4	no	—
280	C₂³	—	2	8	no	—
281	C₂³	—	4	32	no	—
282	C₂³	—	4	16	no	—
283	C₂³	—	4	4	no	—
284	C₂³	—	4	4	no	—
285	C₂³	—	4	16	no	—
286	C₂³	—	8	16	no	—
287	C₂³	—	8	16	no	—
288	C₂³	C₁₃×#120	1	2	yes	—
289	C₂³	C₁₃×#121	1	2	yes	—
290	C₂³	C₁₃×#124	1	2	yes	—
291	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
292	D₄	—	2	8	no	—
293	D₄	—	2	8	no	—
294	D₄	—	4	64	no	—
295	D₄	—	4	16	no	—
296	D₄	—	4	4	no	—
297	D₄	—	4	4	no	—
298	C₂×C₄	—	2	16	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	C₁₃×#120	1	2	yes	—
306	C₂×C₄	C₁₃×#121	1	2	yes	—
307	C₂×C₄	C₁₃×#121	1	8	yes	—
308	C₂×C₄	C₁₃×#123	1	8	yes	—
309	C₂×C₄	C₁₃×#124	1	2	yes	—
310	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	8	yes	—
311	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
313	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
314	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
315	C₈	—	4	4	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	4	8	no	—
320	C₂²	—	4	16	no	—
321	C₂²	—	8	32	no	—
322	C₂²	—	8	8	no	—
323	C₂²	—	8	8	no	—
324	C₂²	C₁₃×#394	1	4	yes	—
325	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
326	C₂²	C₂×C₁₃×#121	1	1	yes	characteristic
327	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
328	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	4	yes	—
329	C₄	—	2	8	no	—
330	C₄	—	4	4	no	—
331	C₄	—	8	2	no	—
332	C₄	C₁₃×#394	1	4	yes	—
333	C₄	C₂×C₁₃×#121	1	2	yes	—
334	C₄	C₂×C₁₃×#123	1	2	yes	—
335	C₂	—	2	4	no	—
336	C₂	—	8	8	no	—
337	C₂	C₂×C₁₃×#394	1	1	yes	characteristic
338	C₂	C₁₃×#3003	1	2	yes	—
339	C₂	C₁₃×#3038	1	2	yes	—
340	C₂	C₁₃×#3138	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×#127	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₁₃×#418	C₂	semidirect
3. C₂×C₁₃×#418	C₂	semidirect
4. C₁₃×#2296	C₂	semidirect
5. C₁₃×#2041	C₂	semidirect
6. C₁₃×#1728	C₂	semidirect
18. C₂×C₁₃×#102	C₂²	semidirect
19. C₂×C₁₃×#102	C₂²	semidirect
20. C₁₃×#389	C₂²	semidirect
23. C₁₃×#336	C₂²	semidirect
46. C₁₃×#117	D₄	semidirect
60. #23505	C₁₃	direct (N × H)
102. C₁₃×C₄⋊C₄	C₂×D₄	semidirect
110. C₂×#418	C₂×C₁₃	semidirect
111. C₂×#418	C₂×C₁₃	semidirect
112. #2296	C₂×C₁₃	semidirect
113. #2041	C₂×C₁₃	semidirect
114. #1728	C₂×C₁₃	semidirect
163. C₂×#102	C₂²×C₁₃	semidirect
164. C₂×#102	C₂²×C₁₃	semidirect
165. #389	C₂²×C₁₃	semidirect
168. #336	C₂²×C₁₃	semidirect
210. #117	D₄×C₁₃	semidirect
271. C₄⋊C₄	C₂×D₄×C₁₃	semidirect
278. C₁₃	#23505	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₃×#127	2
C₂×C₁₃×#418	2
C₁₃×#2296	2
C₁₃×#2041	2
C₁₃×#1728	2
#23505	13

Subgroups — C13×#23505 #15709844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23505 #15709844